Equal Superposition Transformations and Quantum Random Walks

TL;DR

This paper explores equal superposition transformations in qubits, analyzing their properties through various methods, and investigates quantum random walks, revealing how different coin states influence walk symmetry and bias.

Contribution

It characterizes the largest set of qubits satisfying equal superposition transformations and examines the effects of balanced and unbalanced coins on quantum random walks.

Findings

All balanced coins produce identical asymmetric distributions.

Unbalanced coins can create new unbiased walks without classical counterparts.

Multiple methods confirm the maximal ensemble of qubits satisfying the transformation.

Abstract

The largest ensemble of qubits which satisfy the general transformation of equal superposition is obtained by different methods, namely, linearity, no-superluminal signalling and non-increase of entanglement under LOCC. We also consider the associated quantum random walk and show that all unitary balanced coins give the same asymmetric spatial probability distribution. It is further illustrated that unbalanced coins, upon appropriate superposition, lead to new unbiased walks which have no classical analogues.